The Lovász local lemma, as stated in Corollary 5.1.2 here, is given as follows.
Lemma. Let $A_1, \ldots, A_k$ be events such that each $A_i$ has probability at most $p$ and such that each $A_i$ is mutually independent of all but at most $d$ of the $A_j$'s. Then if $ep(d+1) \leq 1$, the probability that none of the $A_i$'s occur is positive.
I have a few questions about the assumptions needed for the above lemma. Looking at the proof given in the Wikipedia article here, it appears that one does not require each $A_i$ to be mutually independent of all but at most $d$ of the $A_j$'s. Instead, all one requires is that$$\text{Prob}\left(A_i \mid \bigwedge_{j \in S} \overline{B_j} \right) = \text{Prob}(A_i),$$where $S = \{1, \ldots, k\} - \Gamma(A_i)$, where $\Gamma(A_i)$ is the dependency locus of $A_i$. I think that the above condition appears to be considerably weaker than mutual independence; indeed, mutual independence means that $$\text{Prob}\left(A_i \wedge \bigwedge_{j \in S} \overline{B_j}\right) = \text{Prob}(A_i) \cdot \prod_{j \in S} \text{Prob}(\overline{B_j}).$$
Am I right to say that mutual independence is not required?
Also, is there a reason why the constant $e$ in the statement of the Lovász local lemma is optimal? Several sources seem to agree on this. It seems somewhat arbitrary to me, and I feel like a smaller constant can be achieved by being more careful in the proof.
Thanks in advance!